Optimal consumption and portfolio in a jump diffusion market

Abstract

We consider the problem of optimal consumption and portfolio in a jump diffusion market consisting of a bank account and a stock, whose price is modeled by a geometric Lévy process. We show that in the absence of transaction costs, the solution in the jump diffusion case has the same form as in the pure diffusion case solved by Merton [M]. In particular, the optimal portfolio is to keep a constant fraction of wealth invested in the stock. This constant is smaller than the corresponding optimal fraction in the pure diffusion case.